0
$\begingroup$

I have a few questions on the topic of extrema and derivatives:

  1. Consider a differentiable function $f:\mathbb{R}\to\mathbb{R}$ that reaches its maximum in $a\in\mathbb{R}$. This implies that $f'(a)=0$.

I think this is true. We now that $f$ is differentiable over $\mathbb{R}$, so $f'(a)$ exists. Because a (global) maximum is always a local maximum, we can conclude that $f'(a)=0$.

  1. If $f \in C^1([0,1])$ reaches its maximum in a point $a\in [0,1]$, then $f'(a)=0$.

This isn't necessarily true when $a=0$ or $a=1$, because then we're not working with an interior point ($f$ is differentiable over $]0,1[$).

Are my answers and way of thinking correct? Thanks for helping!

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.